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Applications of topology to soft matter physics


Project Description

This project will address mathematical problems in the research field of algebraic topology, inspired by the theory of topological defects in soft materials.

In soft matter physics, one often considers materials whose molecules display discrete symmetries. Liquid crystals, for example, typically consist of rod-shaped molecules with no distinguished head or tail. The molecules of such a material tend to align with each other in local patches of the sample region, but there may be places where continuity of alignment breaks down. These so-called topological defects are responsible for interesting behaviours and properties at the mesoscopic scale, and it has long been recognised that tools from algebraic topology (such as fibre bundles and homotopy groups) play a vital role in understanding them.

Biaxial nematics are liquid crystals with more than one axis of reflectional symmetry. One of the goals of this project is to obtain a better understanding of the mathematics behind biaxial nematics and their topological defects. This will involve answering beautiful problems in the topology of manifolds, related to classical topics such as the vector field problem and the Poincaré-Hopf theorem.

Candidates should have (or expect to achieve) a UK honours degree at 2.1 or above (or equivalent) in Mathematics alongside a Masters with Merit or Distinction, also in Mathematics.

It is essential that the successful applicant has a first course in algebraic topology. Knowledge of the basics of differential topology (smooth manifolds, vector bundles, characteristic classes etc) is desirable.

APPLICATION PROCEDURE:

• Apply for Degree of Doctor of Philosophy in Mathematics
• State name of the lead supervisor as the Name of Proposed Supervisor
• State ‘Self-funded’ as Intended Source of Funding
• State the exact project title on the application form

When applying please ensure all required documents are attached:

• All degree certificates and transcripts (Undergraduate AND Postgraduate MSc-officially translated into English where necessary)
• Detailed CV
• Details of 2 academic referees

Informal inquiries can be made to Dr Mark Grant () with a copy of your curriculum vitae and cover letter. All general enquiries should be directed to the Postgraduate Research School ()

Funding Notes

This project is advertised in relation to the research areas of the discipline of Mathematics. The successful applicant will be expected to provide the funding for Tuition fees, living expenses and maintenance. Details of the cost of study can be found by visiting View Website. THERE IS NO FUNDING ATTACHED TO THESE PROJECTS.

References

N.D. Mermin, The topological theory of defects in ordered media, Rev. Mod. Phys. 51 (1979), 591–648.

D. Crowley and M. Grant, The Poincaré-Hopf theorem for line fields revisited, J. Geom. Phys. 117 (2017), 187–196.

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